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Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. (English) Zbl 1049.60091
Extending results of D. Weingarten [J. Math. Phys. 19, 999–1001 (1978; Zbl 0388.28013)], this paper gives explicit formulas for the Haar-average of polynomial functions on the unitary group $$U_d$$. This is applied to probabilistic limit theorems for large $$d$$, in particular to results of asymptotic freeness in quantum probability.

##### MSC:
 60K40 Other physical applications of random processes 81S25 Quantum stochastic calculus 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 43A75 Harmonic analysis on specific compact groups
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